If the plane 2ax - 3ay + 4az + 6 = 0 passes t | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The equation of the first sphere is $x^2 + y^2 + z^2 + 6x - 8y - 2z - 13 = 0$. Its center $C_1$ is $(-3, 4, 1)$.The equation of the second sphere

Vedclass · Accepted Answer

$-2$

Vedclass · Answer

(A) The equation of the first sphere is $x^2 + y^2 + z^2 + 6x - 8y - 2z - 13 = 0$. Its center $C_1$ is $(-3, 4, 1)$.The equation of the second sphere is $x^2 + y^2 + z^2 - 10x + 4y - 2z - 8 = 0$. Its center $C_2$ is $(5, -2, 1)$.The midpoint $P$ of the line segment joining $C_1$ and $C_2$ is given by $P = \left( \frac{-3 + 5}{2}, \frac{4 - 2}{2}, \frac{1 + 1}{2} \right) = (1, 1, 1)$.Since the plane $2ax - 3ay + 4az + 6 = 0$ passes through $P(1, 1, 1)$,we substitute the coordinates of $P$ into the plane equation:$2a(1) - 3a(1) + 4a(1) + 6 = 0$$2a - 3a + 4a + 6 = 0$$3a + 6 = 0$$3a = -6$$a = -2$.

If the plane $2ax - 3ay + 4az + 6 = 0$ passes through the midpoint of the line segment joining the centers of the spheres $x^2 + y^2 + z^2 + 6x - 8y - 2z = 13$ and $x^2 + y^2 + z^2 - 10x + 4y - 2z = 8$,then $a = ......$

Similar Questions

$A$ plane passes through a fixed point $(p, q, r)$ and cuts the axes at $A, B, C$. Then the locus of the centre of the sphere $OABC$ is

What is the center of the sphere passing through the four points $(0, 0, 0), (0, 2, 0), (1, 0, 0),$ and $(0, 0, 4)$?

If the plane $2ax - 3ay + 4az + 6 = 0$ passes through the midpoint of the line joining the centres of the spheres ${x^2} + {y^2} + {z^2} + 6x - 8y - 2z = 13$ and ${x^2} + {y^2} + {z^2} - 10x + 4y - 2z = 8$,then $a$ equals

If $(2, 3, 5)$ is one end of a diameter of the sphere $x^2 + y^2 + z^2 - 6x - 12y - 2z + 20 = 0$,then the coordinates of the other end of the diameter are:

$A$ point moves such that the sum of the squares of its distances from two given points remains constant. The locus of the point is

Vedclass Test Series

Exam Paper Generator

Online Exam Module